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A stable primal–dual approach for linear programming under nondegeneracy assumptions

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Abstract

This paper studies a primal–dual interior/exterior-point path-following approach for linear programming that is motivated on using an iterative solver rather than a direct solver for the search direction. We begin with the usual perturbed primal–dual optimality equations. Under nondegeneracy assumptions, this nonlinear system is well-posed, i.e. it has a nonsingular Jacobian at optimality and is not necessarily ill-conditioned as the iterates approach optimality. Assuming that a basis matrix (easily factorizable and well-conditioned) can be found, we apply a simple preprocessing step to eliminate both the primal and dual feasibility equations. This results in a single bilinear equation that maintains the well-posedness property. Sparsity is maintained. We then apply either a direct solution method or an iterative solver (within an inexact Newton framework) to solve this equation. Since the linearization is well posed, we use affine scaling and do not maintain nonnegativity once we are close enough to the optimum, i.e. we apply a change to a pure Newton step technique. In addition, we correctly identify some of the primal and dual variables that converge to 0 and delete them (purify step).

We test our method with random nondegenerate problems and problems from the Netlib set, and we compare it with the standard Normal Equations NEQ approach. We use a heuristic to find the basis matrix. We show that our method is efficient for large, well-conditioned problems. It is slower than NEQ on ill-conditioned problems, but it yields higher accuracy solutions.

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Authors and Affiliations

  1. Department of Scientific Computing and Statistics, Simon Bolivar University, Caracas, 1080, Venezuela

    Maria Gonzalez-Lima

  2. Department of Combinatorics & Optimization, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

    Hua Wei & Henry Wolkowicz

Authors
  1. Maria Gonzalez-Lima
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  2. Hua Wei
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  3. Henry Wolkowicz
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Correspondence to Henry Wolkowicz.

Additional information

M. Gonzalez-Lima research supported by Universidad Simón Bolívar (DID-GID001) and Conicit (project G97000592), Venezuela. H. Wei research supported by The Natural Sciences and Engineering Research Council of Canada and Bell Canada. H. Wolkowicz research supported by The Natural Sciences and Engineering Research Council of Canada.

URL: http://orion.math.uwaterloo.ca/~hwolkowi/henry/reports/ABSTRACTS.html. This report is a revision of the earlier CORR 2001-66.

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Gonzalez-Lima, M., Wei, H. & Wolkowicz, H. A stable primal–dual approach for linear programming under nondegeneracy assumptions. Comput Optim Appl 44, 213–247 (2009). https://doi.org/10.1007/s10589-007-9157-2

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  • DOI: https://doi.org/10.1007/s10589-007-9157-2

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